What concepts or facts do you know from math that is mind blowing, awesome, or simply fascinating?

Here are some I would like to share:

  • Gödel’s incompleteness theorems: There are some problems in math so difficult that it can never be solved no matter how much time you put into it.
  • Halting problem: It is impossible to write a program that can figure out whether or not any input program loops forever or finishes running. (Undecidablity)

The Busy Beaver function

Now this is the mind blowing one. What is the largest non-infinite number you know? Graham’s Number? TREE(3)? TREE(TREE(3))? This one will beat it easily.

  • The Busy Beaver function produces the fastest growing number that is theoretically possible. These numbers are so large we don’t even know if you can compute the function to get the value even with an infinitely powerful PC.
  • In fact, just the mere act of being able to compute the value would mean solving the hardest problems in mathematics.
  • Σ(1) = 1
  • Σ(4) = 13
  • Σ(6) > 101010101010101010101010101010 (10s are stacked on each other)
  • Σ(17) > Graham’s Number
  • Σ(27) If you can compute this function the Goldbach conjecture is false.
  • Σ(744) If you can compute this function the Riemann hypothesis is false.

Sources:

    • BetoA
      24•1 year ago

      Related: every time you shuffle a deck of cards you get a sequence that has never happened before. The chance of getting a sequence that has occurred is stupidly small.

      • my_hat_stinks
        7•1 year ago

        Most of the time, but only as long as the shuffle is actually random. A perfect riffle shuffle on a brand new deck will get you the same result every time, and 8 perfect riffles on a row get you back to where you started.

    • @dQw4w9WgXcQ@lemm.ee
      5•1 year ago

      I’m guessing this is more pronounced at lower levels. At high level chess, I often hear commentators comparing the moves to their database of games, and it often takes 20-30 moves before they declare that they have now reached a position which has never been reached in a professional game. The high level players have been grinding openings and their counters and the counters to the counters so deeply that a lot of the initial moves can be pretty common.

      Also, high levels means that games are narrowing more towards the “perfect” moves, meaning that repetition from existing games are more likely.

      • @Valmond@lemmy.mindoki.com
        1•1 year ago

        Oh yes I agree totally!

        At those (insane) levels “everyone” knows the “best” move after the choice of opening and for a long time.

        But for the millions of games played every day, I guess it’s a bit less :-)